*3.2. Successive Local Linearization Method*

For the implementation of Successive Local Linearization Method, first we have to reduce the order of Equation (24). To serve the purpose, a new transformation *g* = *h*, leads Equation (10) to Equation (13) into the following form:

$$hh'' + gh' - h^2 - Mh - \beta\_D h + \frac{G\_r}{R\_c^2} (\theta - N\_r \phi - R\_b \phi) \tag{22}$$

$$\frac{1}{P\_r}\theta'' + \theta'[\mathcal{g} + N\_b\phi'] + N\_t\theta r^2 + E\_t\{h\nu^2 + Mh^2\} = 0,\tag{23}$$

$$
\phi'' + L\_t \phi' g + \frac{N\_t}{N\_b} \theta'' = 0,
\tag{24}
$$

$$\left\{\phi''\right.\left. + L\_b g\phi' - P\_t \left\{ [\phi + \Omega\_d] \phi'' + \phi' \phi' \right\} \right.\left. = \left. 0. \right.\tag{25}$$

Linearizing the non-linear term *h*2 by applying Taylor series expansion can be written as

$$h^2 r\_{t+1} = h^2 r + 2h[h\_{t+1} - h\_t] \ = \ 2h\_t h\_{t+1} - h^2\_t \tag{26}$$

where the component having subscripts *t* and *t* + 1 stand for current previous and current approximated values. When Equation (26) is placed in Equation (22), then the non-linear system by means of Gauss-Seidel relaxation method can be decoupled as:

$$\mathbf{g}'\_{t+1} = \mathbf{h}\_t \tag{27}$$

$$\mathbf{h}^{\prime}\mathbf{h}^{\prime}\mathbf{t}\_{t+1} + \mathbf{g}\_{t}\mathbf{h}^{\prime}\mathbf{t}\_{t+1} - \mathbf{M}\mathbf{h}\_{t+1} - \beta\_{\mathrm{D}}\mathbf{h}\_{t+1} - 2\mathbf{h}\_{\mathrm{I}}\mathbf{h}\_{t+1} = \ -\mathbf{h}^{2}\_{\mathrm{I}} - \frac{\mathbf{G}\_{\mathrm{I}}}{R\_{\mathrm{t}}^{2}}(\boldsymbol{\Theta}\_{\mathrm{I}} - \mathbf{N}\_{\mathrm{r}}\boldsymbol{\phi}\_{\mathrm{I}} - \mathbf{R}\_{\mathrm{b}}\boldsymbol{\phi}\_{\mathrm{I}}) \tag{28}$$

$$\frac{1}{P\_r}\theta'\_{t+1} + \theta'\_{t+1}[\underline{g}\_t + N\_b\phi'\_{t}] + N\_t\theta\nu^2\_{t+1} = -E\_c\left\{h\nu^2\_{t+1} + Mh^2\_{t+1}\right\}\tag{29}$$

$$\left[\phi''\!\_{t+1} + \mathrm{L}\_{t}\mathrm{g}\_{t}\phi'\!\_{t+1} + \frac{\mathrm{N}\_{\mathrm{f}}}{\mathrm{N}\_{\mathrm{b}}}\Theta''\!\_{t+1} = \; 0 \tag{30}$$

$$\left[\phi''\_{\ t+1} + L\_b g\_t \phi'\_{\ t+1} - P\_\epsilon([\phi\_{t+1} + \Omega\_d] \phi''\_{\ t+1} + \phi'\_{\ t+1} \phi'\_{\ t+1})\right] = 0\tag{31}$$

The corresponding boundary conditions become

$$g\_{t+1}(0) = 0,\\ h\_{t+1}(0) = 1 = \\\ \theta\_{t+1}(0) = \\\ \phi\_{t+1}(0) = \\\ \phi\_{t+1}(0),\tag{32}$$

$$h\_{t+1}(\infty) = 0 = \theta\_{t+1}(\infty) = \phi\_{t+1}(\infty) = \phi\_{t+1}(\infty),\tag{33}$$

Writing a compact expression of Equations (27)–(31) as follows

$$\mathbf{g}'\_{t+1} = \,^t\!d\_{00} \tag{34}$$

$$h''{}\_{t+1} + d\_{11}h'{}\_{t+1} - d\_{13}h\_{t+1} - 2h\_lh\_{t+1} = \,\_1d\_{1,t} \tag{35}$$

$$\frac{1}{P\_r}\theta''{}\_{t+1} + d\_{11}\theta'{}\_{t+1} + \mathcal{N}\_b \, \phi'{}\_t \theta'{}\_{t+1} + \mathcal{N}\_t \theta \tau^2{}\_{t+1} = d\_{2,t} \tag{36}$$

$$
\phi''|\_{t+1} + d\_{32}\phi'\_{t+1} + \frac{N\_t}{N\_b}\theta''\_{t+1} = d\_{3,t} \tag{37}
$$

$$\left\{ \phi^{\prime\prime}\_{\ \ t+1} + d\_{42} \phi^{\prime}\_{\ \ t+1} - P\_{\epsilon} \Big| [\phi\_{t+1} + \Omega\_d] \phi^{\prime\prime}\_{\ \ t+1} + \phi^{\prime}\_{\ \ t+1} \phi^{\prime}\_{\ \ t+1} \right\} = d\_{4,t} \tag{38}$$

where

$$\begin{array}{rcl}d\_{00} &=& \mathfrak{h}\_{\mathfrak{t}}, d\_{11} &=& \mathfrak{g}\_{\mathfrak{t}\mathfrak{t}}, d\_{12} = 2\mathfrak{h}\_{\mathfrak{t}\mathfrak{t}}, d\_{13} &=& [\mathfrak{M} + \mathfrak{f}\_{\mathbf{D}}], d\_{1,\mathfrak{t}} \\ &=& -\mathfrak{h}\_{\mathfrak{t}}^{2} - \frac{\mathfrak{G}\_{\mathfrak{t}}}{R\_{\mathfrak{t}}^{2}}(\mathfrak{H} - \mathfrak{N}\_{\mathfrak{r}}\mathfrak{g}\_{\mathfrak{t}} - \mathfrak{R}\_{\mathfrak{b}}\mathfrak{g}\_{\mathfrak{t}}) \end{array} \tag{39}$$

$$d\_{2,t} = -\mathbf{E}\_{\mathbf{c}} \Big( \mathbf{h}'^2{}\_{t+1} + \mathbf{M} \mathbf{h}^2{}\_{t+1} \Big), \\ d\_{32} = \mathbf{L}\_{\mathbf{c}} \mathbf{g}\_{t'}, \\ d\_{42} = \mathbf{L}\_{\mathbf{b}} \mathbf{g}\_{t'}, \\ d\_{3,t} = d\_{4,t} = 0 \tag{40}$$

Now, employing the Chebyshev spectral collocation method at the system of Equations (34)–(38), where the differentiation matrix *D* = 2*l D* utilized to perform approximation for the derivatives of unknown variables in the above equations and our new system become

$$D\_{\mathcal{G}t+1} = h\_t \tag{41}$$

$$\left\{\mathbf{D}^2 + \operatorname{diag}[d\_{11}]\mathbf{D} - \operatorname{diag}[d\_{12}]\mathbf{I} - d\_{13}\mathbf{I}\right\} \mathbf{H}\_{\mathbf{f}+1} = d\_{1,\mathbf{f}}\tag{42}$$

$$\left\{\frac{1}{P\_r}\mathbf{D}^2 + \operatorname{diag}[d\_{11}]\mathbf{D} + \operatorname{N}\_b \operatorname{diag}[\boldsymbol{\phi}'\_t]\mathbf{D} + \operatorname{N}\_t\mathbf{D}^2\right\}\boldsymbol{\theta}\_{t+1} = \boldsymbol{d}\_{2,t} \tag{43}$$

$$\left\{\mathbf{D}^2 + \text{diag}[d\_{32}]\mathbf{D} + \frac{N\_t}{N\_b}\text{diag}[\theta''|\_{t+1}]I\right\}\phi\_{t+1} = d\_{3,t} \tag{44}$$

$$\begin{cases} \mathbf{D}^2 + \operatorname{diag}[\mathbf{d}\_{42}] \mathbf{D} - P\_\varepsilon \Omega\_d \operatorname{diag}[\boldsymbol{\phi}^{\prime\prime}\_{\ \ t+1}] \mathbf{I} - P\_\varepsilon \operatorname{diag}[\boldsymbol{\phi}^{\prime\prime}\_{\ \ t+1}] \mathbf{I} \\ \qquad - \operatorname{diag}[\boldsymbol{\phi}^{\prime}\_{\ \ t+1}] \mathbf{D} \end{cases} \middle| \boldsymbol{\phi}\_{t+1} = \boldsymbol{d}\_{4,t} \tag{45}$$

With their respective boundary conditions

$$g\_{t+1}(\eta\_N) = 0, h\_{t+1}(\eta\_N) = 1 \ = \theta\_{t+1}(\eta\_N) = \phi\_{t+1}(\eta\_N) \ = \phi\_{t+1}(\eta\_N) \tag{46}$$

$$h\_{t+1}(\eta\_0) = 0 \ = \ \theta\_{t+1}(\eta\_0) = \phi\_{t+1}(\eta\_0) = \phi\_{t+1}(\eta\_0),\tag{47}$$

The system can be expressed in a more simplified way as

$$B\_1 \mathbb{S}\_{t+1} = E\_1 \tag{48}$$

$$B\_2 h\_{t+1} = E\_2 \tag{49}$$

$$B\_3 \theta\_{t+1} = E\_3 \tag{50}$$

$$B\_4 \phi\_{t+1} = E\_4 \tag{51}$$

$$B\mathfrak{g}\phi\_{t+1} = E\mathfrak{z} \tag{52}$$

where

$$B\_1 = D\_\prime E\_1 = \hbar\_\prime \tag{53}$$

$$B\_2 = D^2 + \text{diag}[d\_{11}]D - \text{diag}[d\_{12}]I - d\_{13}I,\\ E\_2 = d\_{1,t} \tag{54}$$

$$B\_3 = \frac{1}{P\_r} D^2 + \operatorname{diag} [d\_{11}] D + \operatorname{N}\_b \operatorname{diag} [\phi'\_t] D + \operatorname{N}\_l D^2, E\_3 = d\_{2,t} \tag{55}$$

$$B\_4 = D^2 + \text{diag}[d\_{32}]D + \frac{\text{N}\_\text{t}}{\text{N}\_\text{b}} \text{diag}[\theta^{\prime\prime}\_{\text{ t}+1}]I\_\text{}^\prime E\_4 = d\_{3,t\prime} \tag{56}$$

$$\begin{array}{rcl} B\_5 &=& D^2 + \operatorname{diag}[\mathbf{d}\_{42}]D - P\_t \Omega\_d \operatorname{diag}[\boldsymbol{\phi}^{\prime\prime}\boldsymbol{\iota}\_{t+1}]\mathbf{I} - P\_t \operatorname{diag}[\boldsymbol{\phi}^{\prime\prime}\boldsymbol{\iota}\_{t+1}]\mathbf{I} \\ &- \operatorname{diag}[\boldsymbol{\phi}^{\prime}\boldsymbol{\iota}\_{t+1}]\mathbf{D}\_\cdot E\_5 &=& d\_{4,t} \end{array} \tag{57}$$

$$\operatorname{diag}[d\_{\mathrm{I1}}] = \left[ \begin{array}{cccc} d\_{\mathrm{I1}}(\eta\_{\mathrm{0}}) & \cdots & & \\ \vdots & \ddots & \vdots \\ & & \cdots & d\_{\mathrm{I1}}(\eta\_{\mathrm{N}}) \end{array} \right] \operatorname{diag}[d\_{\mathrm{I2}}] = \left[ \begin{array}{cccc} d\_{\mathrm{I2}}(\eta\_{\mathrm{0}}) & \cdots & & \\ \vdots & \ddots & \vdots \\ & & \cdots & d\_{\mathrm{I2}}(\eta\_{\mathrm{N}}) \end{array} \right] \tag{58}$$

$$\operatorname{diag} [d\_{1,t}] = \begin{bmatrix} d\_{1,t}(\eta\_0) \\ \vdots \\ d\_{1,t}(\eta\_N) \end{bmatrix} \\ \operatorname{diag} [d\_{2,t}] = \begin{bmatrix} d\_{2,t}(\eta\_0) \\ \vdots \\ d\_{2,t}(\eta\_N) \end{bmatrix} \tag{59}$$

$$\operatorname{diag}[d\_{32}] = \left| \begin{array}{ccccc} d\_{32}(\eta\_0) & \cdots & & \\ \vdots & \ddots & \vdots \\ & \cdots & d\_{32}(\eta\_N) \end{array} \right| \operatorname{diag}[d\_{42}] = \left| \begin{array}{ccccc} d\_{42}(\eta\_0) & \cdots & & \\ \vdots & \ddots & \vdots \\ & \cdots & d\_{42}(\eta\_N) \end{array} \right| \right| \tag{60}$$

$$d\_{3,t} = d\_{4,t} = \begin{array}{c} 0 \\ \vdots \\ 0 \end{array} \Big| \tag{61}$$

$$g\_{t+1} = \begin{bmatrix} \mathbf{g}(\eta\_0), \mathbf{g}(\eta\_1), \dots, \mathbf{g}(\eta\_{\mathcal{N}}) \end{bmatrix}^T,\\ h\_{t+1} = \begin{bmatrix} h(\eta\_0), h(\eta\_1), \dots, h(\eta\_{\mathcal{N}}) \end{bmatrix}^T,\tag{62}$$

$$\theta\_{t+1} = \begin{bmatrix} \theta(\eta\_0), \theta(\eta\_1), \dots, \theta(\eta\_N) \end{bmatrix}^T,\\ \phi\_{t+1} = \begin{bmatrix} \phi(\eta\_0), \phi(\eta\_1), \dots, \phi(\eta\_N) \end{bmatrix}^T \tag{63}$$

φ*t*+<sup>1</sup> = [φ(η0), φ(η1), ... , φ(η*N*)]*<sup>T</sup>* are vectors of sizes (*N* + 1) × 1 whereas 0 is a vector of order (*N* + 1) × 1 and *I* is an identity matrix of order (*N* + 1) × (*N* + <sup>1</sup>).

In view of boundary conditions, the Equations (48)–(63) take the following form:

$$\begin{aligned} \begin{array}{c} \begin{array}{c} \\ \begin{array}{c} \\ \begin{array}{c} \\ \begin{array}{c} \\ \end{array} \end{array} \end{array} \left| \begin{array}{c} \begin{array}{c} \mathcal{G}\_{l+1}(\eta\_{l}) \\\\ \begin{array}{c} \\ \end{array} \end{array} \right| \begin{array}{c} \mathcal{G}\_{l+1}(\eta\_{l}) \\\\ \begin{array}{c} \\ \end{array} \end{array} \right| \begin{array}{c} \\\\ E\_{l} = \begin{array}{c} \\\\ \end{array} \end{array} \left| \begin{array}{c} E\_{1} \\\\ \begin{array}{c} \\ \begin{array}{c} \\ \end{array} \end{array} \right| \begin{array}{c} \\\\ \begin{array}{c} \\ \end{array} \end{array} \right| \begin{array}{c} \\\\ \begin{array}{c} \\ \end{array} \end{array} \left| \begin{array}{c} \\\\ \begin{array}{c} \\ \end{array} \right| \begin{array}{c} \\\\ \end{array} \right| \begin{array}{c} \\\\ \end{array} \end{array} \right| \begin{array}{c} \\\\ \begin{array}{c} \\\\ \end{array} \end{aligned} \end{aligned} \end{aligned} \tag{64}$$

$$\begin{aligned} \;^1E\_2 = \left[ \begin{array}{c} \frac{\overline{0}}{E\_2} \\\\ \overline{1} \end{array} \right] \;^B\_3 = \left[ \begin{array}{c} \frac{1 \ \dots \ 0}{B\_3} \\\\ \hline 0 \ \dots \ 1 \end{array} \right] \;^C\_{t+1} = \left[ \begin{array}{c} \frac{\theta\_{t+1}(\eta\_0)}{\theta\_{t+1}(\eta\_1)} \\\\ \vdots \\\\ \frac{\theta\_{t+1}(\eta\_N)}{\theta\_{t+1}(\eta\_N)} \end{array} \right] \;^E\_3 = \left[ \begin{array}{c} \frac{\overline{0}}{E\_3} \\\\ \overline{1} \end{array} \right] \end{aligned} \tag{65}$$

$$\left[\underbrace{\begin{array}{c} \mathbf{1} \\ \hline \hline \mathbf{1} \\ \hline \hline \mathbf{1}\_{\mathsf{A}\_{4}} \\ \hline \hline \mathbf{0} \\ \hline \end{array}}\_{\in \Phi \rightarrow 1} \left[\underbrace{\begin{array}{c} \phi\_{1+1}(\eta\_{1}) \\ \phi\_{1+1}(\eta\_{1}) \\ \hline \hline \end{array}}\_{\in \Phi \rightarrow 1} \left[\underbrace{\begin{array}{c} \mathbf{2} \\ \hline \hline \mathbf{1}\_{\mathsf{A}\_{4}} \\ \hline \end{array}}\_{\in \Phi} \left[\underbrace{\begin{array}{c} \mathbf{1} \\ \hline \hline \mathbf{2}\_{5} \\ \hline \hline \end{array}}\_{\in \Phi \rightarrow 1} \left[\underbrace{\begin{array}{c} \phi\_{1+1}(\eta\_{1}) \\ \hline \hline \phi\_{1+1}(\eta\_{1}) \\ \hline \hline \end{array}}\_{\in \Phi \rightarrow 1} \left[\underbrace{\begin{array}{c} \mathbf{3} \\ \hline \hline \mathbf{2}\_{5} \\ \hline \end{array}}\_{\in \Phi} \end{array}\_{\in \Phi \rightarrow 1} \left[\underbrace{\begin{array}{c} \mathbf{4} \\ \hline \hline \mathbf{5} \\ \hline \end{array}}\_{\in \Phi} \left[\mathbf{5} \right] \right] \right] \right]$$

The initial guesses are:

$$g\_0(\eta) = (1 - e^{-\eta}), \; h\_0(\eta) = e^{-\eta}, \; \theta\_0(\eta) = \phi\_0(\eta) = \phi\_0(\eta) = e^{-\eta} \tag{67}$$

These initial assumptions approximation satisfying the boundary conditions (46)–(47) achieve subsequent approximations of *gt*, *ht*, θ*<sup>t</sup>*,φ*<sup>t</sup>*,φ*<sup>t</sup>* for each *t* = 1, 2, ...... by employing the successive local linearization method.
